37 research outputs found
Minimizing the Euclidean Condition Number
This paper considers the problem of determining the row and/or column scaling of a matrix A that minimizes the condition number of the scaled matrix. This problem has been studied by many authors. For the cases of the ∞-norm and the 1-norm, the scaling problem was completely solved in the 1960s. It is the Euclidean norm case that has widespread application in robust control analyses. For example, it is used for integral controllability tests based on steady-state information, for the selection of sensors and actuators based on dynamic information, and for studying the sensitivity of stability to uncertainty in control systems.
Minimizing the scaled Euclidean condition number has been an open question—researchers proposed approaches to solving the problem numerically, but none of the proposed numerical approaches guaranteed convergence to the true minimum. This paper provides a convex optimization procedure to determine the scalings that minimize the Euclidean condition number. This optimization can be solved in polynomial-time with off-the-shelf software
Generalized Perron--Frobenius Theorem for Nonsquare Matrices
The celebrated Perron--Frobenius (PF) theorem is stated for irreducible
nonnegative square matrices, and provides a simple characterization of their
eigenvectors and eigenvalues. The importance of this theorem stems from the
fact that eigenvalue problems on such matrices arise in many fields of science
and engineering, including dynamical systems theory, economics, statistics and
optimization. However, many real-life scenarios give rise to nonsquare
matrices. A natural question is whether the PF Theorem (along with its
applications) can be generalized to a nonsquare setting. Our paper provides a
generalization of the PF Theorem to nonsquare matrices. The extension can be
interpreted as representing client-server systems with additional degrees of
freedom, where each client may choose between multiple servers that can
cooperate in serving it (while potentially interfering with other clients).
This formulation is motivated by applications to power control in wireless
networks, economics and others, all of which extend known examples for the use
of the original PF Theorem.
We show that the option of cooperation between servers does not improve the
situation, in the sense that in the optimal solution no cooperation is needed,
and only one server needs to serve each client. Hence, the additional power of
having several potential servers per client translates into \emph{choosing} the
best single server and not into \emph{sharing} the load between the servers in
some way, as one might have expected.
The two main contributions of the paper are (i) a generalized PF Theorem that
characterizes the optimal solution for a non-convex nonsquare problem, and (ii)
an algorithm for finding the optimal solution in polynomial time
Solving singular generalized eigenvalue problems. Part II: projection and augmentation
Generalized eigenvalue problems involving a singular pencil may be very
challenging to solve, both with respect to accuracy and efficiency. While Part
I presented a rank-completing addition to a singular pencil, we now develop two
alternative methods. The first technique is based on a projection onto
subspaces with dimension equal to the normal rank of the pencil while the
second approach exploits an augmented matrix pencil. The projection approach
seems to be the most attractive version for generic singular pencils because of
its efficiency, while the augmented pencil approach may be suitable for
applications where a linear system with the augmented pencil can be solved
efficiently
On eigenvalues of rectangular matrices
Given a -tuple of -matrices with we call the set of all -tuples of complex numbers \{\la_1,...,\la_k\}
such that the linear combination A+\la_1B_1+\la_2B_2+...+\la_kB_k has rank
smaller than the {\it eigenvalue locus} of the latter pencil. Motivated
primarily by applications to multi-parameter generalizations of the
Heine-Stieltjes spectral problem, see \cite{He} and \cite{Vol}, we study a
number of properties of the eigenvalue locus in the most important case
.Comment: 10 pages, no figures, LaTeX2